Calculus 3 Lecture 13.2: Limits and Continuity of Multivariable Functions (with Squeeze Th.)
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- Опубліковано 19 вер 2024
- Calculus 3 Lecture 13.2: Limits and Continuity of Multivariable Functions: How to show a limit exits or Does Not Exist for Multivariable Functions (including Squeeze Theorem). Also, how to determine regions of continuity.
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Intro to multivariable limits: 0:00
Showing a limit (with two variables) does not exist: 26:54
Showing a limit (with three variables) does not exist: 1:12:19
Showing a limit does exist: 1:32:07
changing to polar coordinates: 1:32:28
Continuity (domain and compositions): 1:41:47
Summary of computing limits: 2:06:56 - 2:07:24
Squeeze theorem (converting to inequalities): 2:07:26
♈🔧
Where’s the balloon
Thanks a lot for the timestamps!
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conceptual talk until 00:00
Examples of 2 dependent variables: 24:00
Examples of 3 dependent variables: 1:12:15
Squeeze theorem : 2:07:25
Independent*
thanks
Elite
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watching this 8 hours before the exam
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Thank you for showing how to apply Squeeze Theorem to these problems!!! My teacher tried to explain it to us in half of a lecture and you did it under 10 minutes in a way that I can understand!
My professor for Calc 3 is named Hongwei and speaks with a hard accept from china so when he goes over anything its always a bigger challenge to not only understand the concept but him as well. So when he went over this section he sped through it only talking about the epsilon-delta definition of it and then using polar to solve some problems, never fully explaining the actual concept. This video has not only saved my grade but also my understanding of limits for all future calculations I'll be doing in my career as an engineer. Thank you so much Professor Leonard!
I have been struggling with limit continuities for weeks, this video's first ten minutes cleared up everything I had wrong!
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It's funny how on calc 1, I watched your videos in regular speed. In calc 2 it was in 2x it's speed, and now I'm able to manually watch calc 3 in 3x it's speed. Just goes to show how clear it is, thanks so much for creating smooth transitions to new topics and info (:
Professor Leonard,
May god bless you with a continuous eternal function of health and wealth with no holes and asymptotes ☺😘💝
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Your classes are helping me get through this multivariable stuff during these trying times. Thank you!
Im from Argentina and this is the best video i found to understand limits, ty man the only one who say that the point have to be on the path, sorry for my english.
Professor Leonard, I'm always excited to see new material. I'm currently in Calc 3, and you've been a major help along the way! I have finals next week and I can understand your probably very busy yourself... Do you plan to release more videos in the next week or so? Thanks!
+Brad Nissen I'm releasing them as fast as I can make them, I hope they will help. Good luck!
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He actually already has released it, but the video was shot with a 4-D device in 4-D setting so unfortunately 3-D people are unable to see it. Maybe you should get 4-D glasses, I am selling if you are interested.
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Could you also give more examples of proving the limit DOES exist in different ways and what methods to use when?
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Professor Leonard, thank you for another well organized video/lecture on Limits and Continuity on Multivariable Functions with the classic Squeeze Theorem in Calculus Three. These concepts are introduced in Calculus One, which improve my understanding of the subject in Multivariable Calculus.
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Here too... (1:18:48)
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i think you might be wrong here. When you use for example polar coordinates and you get a number it's not strong enough to prove that the limit exists, you'd have to use that number as a candidate of a posible value of the limit and then plug it the squeeze theorem or the epsilon-delta definition.
Then just then you can prove the limit exists.
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In this first and second example if you do a 3D plot (Used Mathematica in my case) you can see that these functions are not continuous at (0,0) but has a cusp along one axis at the point (0,0) in example 1 and a cusp along the -x,y and x.y diagonals in example 2 therefore the limit is undefined. In the example lim as (x,y)->(1,0) of (2xy-2y)/(x^2 +y^2 -2x +1)Professor Leonard found that the lim as y->0 of 2y^2/2y^2 =0 presumably because this reduced to the lim y->0 of 2y^2/2y^2 is 0/0 =1 and concluded that the limit was undefined. However Mathematica finds this limit of the base limite to be 0, so the original problem has a limit of 0 also which is also what Mathematica found. This is in agreement with the plot of the surface and examining the point (x,y) = (1,0) Mathematica's determination of the original limit to be 0, so I believe his answer is differant. I guess this depends on your definition of 0/0 or it could be a bug in Mathematica. Please comment Professor Leonard and others if you can. Time point 1:11:48.
Thank you so much! I'm surprised my book never mentions the squeeze theorem and instead goes through a bunch of other non-sensical stuff... You made it all perfectly clear, and the theorem kind of makes sense intuitively as well.
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Professor Leonard, Thank you for your videos they have been a big help to supplement my course work. Does your class deal with epsilon-delta definition to prove limits of multi-variable functions?
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I wanted to know what if the question asks to prove that limit exists...
Hey bud, We can always use the squeeze theorem to prove that a limit exists. Refer to 2:07:00 in the video
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@1:36:45 someone asked why r approaches 0 from the positive side. r is the radius of the parameterized curve, radius is a distance , so always positive and that's why approaching from the positive side ✌
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